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BIG PHYSICS, BIG QUESTIONS –

Curious mathematical law is rife in nature

By Rachel Courtland

WHAT do earthquakes, spinning stellar remnants, bright space objects and a host of other natural phenomena have in common? Some of their properties conform to a curious and little known mathematical law, which could now find new uses.

A subject of fascination to mathematicians, Benford’s law states that for many sets of numbers, the first or “leading” digit of each number is not random. Instead, there is a 30.1 per cent chance that a number’s leading digit is a 1. Progressively higher leading digits get increasingly unlikely, and a number has just a 4.6 per cent chance of beginning with a 9 (see diagram).

The law is named after physicist Frank Benford, who in 1938 showed that the trend appears in many number sets, from the surface area of rivers to baseball statistics to figures picked randomly from a newspaper. It later emerged that such distributions are “scale-invariant”&colon; if you convert the units of the numbers in the set, from metres to yards, say, the set will still conform to Benford’s law.

Not all sets of numbers obey this law, but it crops up surprisingly often. As a result, mathematicians have put it to work, using deviations from it to detect cases of tax fraud, voter fraud and even digital image manipulation.

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Now Malcolm Sambridge of the Australian National University in Canberra and colleagues have extended the list of natural phenomena with properties that follow Benford’s law. Their new list includes the depths of almost 250,000 earthquakes that occurred worldwide between 1989 and 2009, the brightness of gamma rays that reach Earth as recorded by the Fermi space telescope, the rotation rates of spinning star remnants known as pulsars, and 987 infectious disease numbers reported to the World Health Organization in 2007 (Geophysical Research Letters, DOI&colon; 10.1029/2010GL044830).

It applies to quakes, the brightness of gamma rays reaching Earth and the rotations of dead stars

That Benford’s law pops up in so many natural phenomena won’t surprise mathematicians but may shock some scientists. When Sambridge’s team presented Benford’s law findings at a 2009 geoscience conference, one dubious attendee “thought we were having a laugh”, he recalls.

Yet geoscience is ripe for new uses of the law, he says. As well as measuring earthquake depths, Sambridge’s team also looked at the vertical displacements of the ground in Peru as the tsunami-triggering Sumatra-Andaman earthquake of 2004 progressed. A set of ground shifts before the earthquake proper, when small movements were due to “background noise”, did not follow Benford’s law, but shifts that occurred during the quake itself did.

The team also examined seismic data recorded at the same time by a station in Canberra. The overall patterns in the shifts persisted but the exact extent of the adherence to the law varied differently over time than in the Peruvian measurements. The team then looked more closely at Canberra seismograms and found that they were consistent with a minor, local earthquake occurring at the same time, which could be the source of the discrepancy between the two measurements.

“That’s the first time I know of where something physical like that was actually discovered using Benford’s law,” says Theodore Hill, a mathematician at the Georgia Institute of Technology in Atlanta, not involved with the work.

As well as using Benford’s law to detect mild earthquakes, Sambridge says it could find other uses. “I’m hoping people will check it out in their data. It could signal something strange and something to investigate, perhaps something that you might not have spotted in another way.” And checking if properties that adhere to Benford’s law in nature also do so in computer simulations could be a way to check and improve misbehaving models.

Just how widespread the law is in nature is not known. When the team looked at the masses of 400 extrasolar planets, there was an anomalous bump in numbers starting with 6. This may be an artefact of a small sample, a problem with the measurement technique or a sign that exoplanet masses do not fit Benford’s law.